This topological characterization has physical implications both in the bulk and at the boundary of the quasiperiodic system: (i) moving from one quasiperiodic modulation \(\tau _1\)  to another \(\tau _2\)  will induce rearrangements of the bulk gaps (phase transitions), at energy scales corresponding to the approximant at which the two modulation separate from one another [1]}; (ii) at any periodic approximant level the topological phase transitions will be avoided with a correction of \(O(1/q_n)\)  [2]}, [3]}, [4]}; (iii) the above two points manifest at a smoothly-interpolating spatial boundary; there, subgap boundary modes with increasing density appear as a function of the boundary length [5]}; (iv) the filling-factor gap labels can appear in bulk diffraction experiments [6]}; (v) the relation to the topological pump implies that at a finite boundary to the vacuum, the bulk gap will be densely filled by subgap modes as a function of \(\phi \)  or equivalently, as a function of removing the last site of a half-infinite chain [2]}, [8]}, [4]}, [10]}, [11]}, which can also be understood as a result of the mechanical work associated with the bulk Chern density, manifesting at the boundary [12]}; and (vi) at higher dimensions \(D>1\) , the boundary manifests a variety of effects with co-dimension up to \(D\) , corresponding to bulk responses of higher Chern numbers [13]}, [14]}, [15]}, [16]}.
